论文标题
小球上点上的覆盖物的注释
A note on point-finite coverings by balls
论文作者
论文摘要
我们提供了V.P. 〜fonf和C.〜zanco结果的基本证明，这是可分离的希尔伯特空间的点覆盖物。确实，通过使用J.〜lindenstrauss和R.R.〜Phelps \ cite \ cite {lp}引入的著名论点的变化来证明反身无限二维Banach空间的单位球无需无数的极端点，我们证明了以下结果：让$ x $ x $满足$ x $满足的hilbert hilbert Space满足。 $ \ mathrm {dens}（x）<2^{\ aleph_0} $，然后$ x $不通过开放或封闭的球（每个正半径）接收点填充覆盖点。 在本文的第二部分中，我们遵循V.P.介绍的论点。 Fonf，M。Levin和C. Zanco在\ cite {fonflevzan14}中证明，先前的结果也存在于无限二维的Banach空间中，这些空间既均匀均匀且均匀地光滑。
We provide an elementary proof of a result by V.P.~Fonf and C.~Zanco on point-finite coverings of separable Hilbert spaces. Indeed, by using a variation of the famous argument introduced by J.~Lindenstrauss and R.R.~Phelps \cite{LP} to prove that the unit ball of a reflexive infinite-dimensional Banach space has uncountably many extreme points, we prove the following result: Let $X$ be an infinite-dimensional Hilbert space satisfying $\mathrm{dens}(X)<2^{\aleph_0}$, then $X$ does not admit point-finite coverings by open or closed balls, each of positive radius. In the second part of the paper, we follow the argument introduced by V.P. Fonf, M. Levin, and C. Zanco in \cite{FonfLevZan14} to prove that the previous result holds also in infinite-dimensional Banach spaces that are both uniformly rotund and uniformly smooth.
